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I am thinking of the following situation:

The lie algebra $g = sl_2(\mathbb(C))$, and $V(1)$ is the unique dimension 2 irreducible representation (the defining representation). Let $U$ be the universal enveloping algebra of $g$.

I pick a basis $v_1, v_2$ for $V(1)$, where $H v_1 = v_1$ and $Hv_2 = -v_2$, and $v^1, v^2$ is the dual basis. (So $Ev_1 = 0$, $Fv_2 = 0$, $E v_2= v_1$, $Fv_1 = v_2$.)

$c^i_j(u) = v^i( u. v_j)$, for $u \in U$, is a matrix coefficient for the Hopf-algebra $H$.

I need to verify that $c^1_1 c^2_2 - c^1_2 c^2_1 = 1$. (In order to show that the algebra of matrix coefficients is isomorphic to the coordinate ring of $SL_2(\mathbb{C})$.)

Here is my attempt:

$c^1_1 c^2_2(u) = m \circ (v^1 \otimes v^2) [ \Delta(u) (v_1 \otimes v_2)] = m \circ (v^1 \otimes v^2)[ ( u \otimes 1 + 1 \otimes u). (v_1 \otimes v_2)] = v^1(uv_1) v^2(v_2) + v^1(v_1) v^2 (u.v_2) = v^1( u. v_1) + v^2 (u. v_2)$

Plugging in $u = E$, $F$ or $H$ from the Lie algebra gives $0$, given my choice of basis above. Plugging in $1$ gives $2$, so $c^1_1 c^2_2 = 2$. (By which I mean, 2 times the unit of $U^*$.)

$c^1_1 c^2_2(u) = m \circ (v^1 \otimes v^2) [ \Delta(u) (v_2 \otimes v_1)] = m \circ (v^1 \otimes v^2)[ ( u \otimes 1 + 1 \otimes u). (v_2 \otimes v_1)] = v^1(uv_2) v^2(v_1) + v^1(v_2) v^2 (u.v_2) = 0$

(since $v^i(v_j) = \delta^i_j$).

Clearly, the difference is not $1$! Please tell me what I am doing wrong.

Thank you for reading!

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  • $\begingroup$ I tried to calculate this using the result from one of your previous questions but got the same result; I also see no error in your calculations. (But I don’t know if these really are the right calculations to make.) But I would like to point out that plugging in $1$ should not give you $2$: Notice that $\Delta(1) = 1 \otimes 1 \neq 1 \otimes 1 + 1 \otimes 1$, as $\Delta$ is an algebra morphism. Instead we get $(c^1_1 c^2_2)(1) = v^1(v_1)v^2(v_2) = 1$. $\endgroup$ – Jendrik Stelzner Jan 24 '16 at 5:02
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    $\begingroup$ @JendrikStelzner Oh, you are right about the image of $1$ Thank you - I see that what I computed was for $u \in g$, so I was mistaken to plug in $1$. I think this answers my question - now I have to show that $c_1^1 c^2_2 - c^1_2 c^2_1$ is an algebra homomorphism to conclude. Background: I am considering the map from $\mathbb{C}[a,b,c,d] \to O$, where $O$ is the matrix coefficient algebra, sending $(a,b,c,d) \to (c^1_1, c^1_2, c^2_1, c^2_2)$. I have already shown this is surjective, and now I just need to show that after this relation $ad - bc = 1$ is taken care of, the map becomes injective. $\endgroup$ – Lorenzo Najt Jan 24 '16 at 5:07
  • $\begingroup$ (It ended up being substantially easier to show directly that this map agreed with the unit on "higher order" pure "tensors" than to show first that it was an algebra homomorphism.) $\endgroup$ – Lorenzo Najt Jan 24 '16 at 16:35

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